High Bandwidth AFM Cantilevers for Operation in Air or Vacuum
In comparison with other AFM (atomic force microscopy) imaging modes, increasing the imaging speed of AFM in dynamic modes in air or vacuum has proven especially difficult. These dynamic modes include non-contact AFM, intermittent contact AFM (tapping mode), and pulsed force or peak force modes. The reason for this difficulty is the slow dynamic response of the cantilever oscillation amplitude. When subject to a change in boundary condition, the transient response of the cantilever decays with a time constant related to its resonance frequency, f0, and quality factor, Q. This time constant directly impacts the imaging bandwidth, B, of the cantilever, which is a measure of how quickly the AFM cantilever can track surface topography changes. For example, in tapping mode, in the case of a linear tip sample interaction and a cantilever driven at resonance, B takes the numerical value B=πf0/Q.
The row marked “Low f0, high Q” of FIG. 1 shows on the left a schematic of a driven oscillating cantilever with a steady state oscillation amplitude subject to a sudden increase in tip-sample distance. After some time, the cantilever reaches a new steady-state oscillation amplitude according to the new boundary conditions. The response time is linked to the cantilever resonance frequency and Q-factor, which are represented in the amplitude versus frequency drawing. Immediately to the right, we see experimental data of the deflection response time of a typical tapping-mode-in-air AFM cantilever (RTESPA, Bruker AFM Probes) subject to a sudden step increase in the drive amplitude. The corresponding resonance frequency and quality factor were measured for this cantilever with the thermal tune method. The numerical values, f0=347 kHz and Q=500, yield a response time of order one millisecond, and a corresponding imaging bandwidth of order 1 kHz. Finally, on the right, we see an SEM image of the cantilever showing its dimensions.
Thus far, efforts to increase the cantilever bandwidth have focused on increasing the resonance frequency by reducing the cantilever dimensions (so called ‘small cantilevers’). This approach, shown in the row marked “High f0, high Q” in FIG. 1, has by and large been the enabling technology for state of the art high-speed AFM, producing good-quality, high-speed images even on difficult biological samples.
The higher resonance frequency reduces the response time, even with largely unchanged Q, as shown schematically on the left and experimentally on the right of the second row in FIG. 1 using a commercially-available small cantilever (FastScan A, Bruker AFM Probes). Cantilevers with widths approaching the optical diffraction limit and lengths of a few micrometres are now usable in specialized high-speed AFMs.
The fastest AFM imaging in dynamic modes has been, however, uniformly performed in a liquid environment, where the Q of the cantilever is decreased substantially by the large amount of fluid damping in liquid (in fluid, most AFM cantilevers have Q≈3). However, reaching equivalent speed performance using dynamic modes in air or vacuum, where the fluid damping is substantially lower, has yet to be shown.
The work of the inventors leading to this invention has focused on an alternate cantilever construction approach to enable cantilevers with inherently low Q. Intrinsically, Q is related to the damping of the resonator. Sources of damping include fluid (air or water), mechanical clamping losses, and internal friction—which includes both surface effects and volume effects such as thermoelastic damping or viscoelastic damping. Each of these sources contribute to the overall Q of the system, which can be expressed as a combination of the Q, associated with each individual damping source:
      1    Q    =            ∑              +                  Q          i                      =                  1                  Q          medium                    +              1                  Q          material                    +              1                  Q          support                    +      …      
Optimizing the cantilever bandwidth through Q reduction therefore translates in practice to increasing the damping mechanisms present in the system.
AFM cantilevers that are commercially available are made out of materials with low intrinsic damping, such as crystalline silicon or silicon nitride. The Intrinsic damping coefficient (or loss coefficient) of a material ηi is defined as the ratio of the imaginary component of the dynamic modulus E″ to the real component of the dynamic modulus E′,
      η    i    =                    E        ″                    E        ′              .  FIG. 2 shows some potential cantilever materials classified by their intrinsic damping coefficient on the horizontal axis, and the square root of the ratio of the elastic modulus E to the density ρ, √(E/ρ), on the vertical axis. This factor E/ρ is also known as the stiffness to weight ratio. Here, the square root in this ratio √(E/ρ) comes from the expression for the fundamental resonance frequency of a cantilever beam
      f    0    =            0.56              l        2              ⁢                  I        A              ⁢                  E        ρ            factored into geometrical and materials properties terms. In the expression, l is the cantilever length, I is the second moment of area and A is the cross-sectional area of the cantilever beam. Three different classes of materials are given: crystalline or ceramic materials, metals, and polymers or elastomers. The dashed lines show constant values of the product ηi√(E/ρ), which a measure of the bandwidth ratio f0/Q expressed in terms of materials properties. Higher values trend towards the upper left corner of the plot. By this metric, the polymers and elastomers as a class of material are roughly 3 orders of magnitude better than the crystals and ceramics.
For cantilevers made of materials with very low damping coefficient, such as silicon or silicon nitride, the primary damping source is thus the air damping, which leads to cantilever with Q≈500 in air as shown in the row marked “Low f0, high Q” of FIG. 1.
If the cantilever is made out of a material that exhibits large intrinsic damping, the material damping dominates the cantilever response. The total number of oscillation cycles needed to reach steady state in this case is decreased, and so the response time is decreased, even though the frequency of oscillation may be unchanged.
Of the various classes of materials that can be used to microfabricate cantilevers, polymers in particular can exhibit large intrinsic damping through viscoelastic effects, making them ideal materials for this application. Shown in the row “Low f0, low Q” in FIG. 1 is a cantilever designed to maintain similar resonance frequency and spring constant as a standard tapping mode in air cantilever, but with a drastically lower Q. The cantilever is made out of a photosensitive polymer (SU-8), which has previously been demonstrated as a promising low-cost, low spring constant cantilever microfabrication material for both AFM imaging and cantilever-based biosensing. The experimental data show that the cantilever, which has a Q=21, has a similar response to a change in boundary condition to the commercial small cantilever (row marked “High f0, high Q” of FIG. 1), even though the size and resonance frequency of the SU-8 cantilever is comparable to a standard cantilever (and is useable in standard AFM systems).
These two approaches, high resonance frequency and low quality factor, can be combined by reducing the dimensions of the cantilever and making a small cantilever out of a viscoelastic material. The row marked “High f0, low Q” in FIG. 1 illustrates this combination. The resulting cantilever, which has a resonance frequency similar to the commercial small cantilever and a Q similar to the large SU-8 cantilever, has a response time that appears almost instantaneous on the timescale shown.
The inventors performed high-speed AFM imaging using these small SU-8 cantilevers in a customized AFM for high-speed operation. The scratched mica surface we imaged has sharp step edges, which are a difficult feature for topography feedback.
FIG. 3a shows images of the same imaging area at line scan rates from 43 Hz up to 166 Hz. The highest scan rate corresponds to scan speeds in excess of 1 image/s and a linear tip speed over the surface of 1953 μm/s. The height images (left of FIG. 3a) appear identical at all scan rates, apart from more pixelation at higher speed due to system data-rate limits. Notable in the amplitude error images (right of FIG. 3a) is the absence of imaging artefacts such as overshoot, parachuting or ringing. Taken together, these images show good tracking behaviour even at the highest scan rates.
Faster AFM imaging also enables the ability to take a high-resolution overview image and digitally investigate regions at higher magnification. This task presents a challenge to AFM systems because at large scan areas, the surface speed remains high even for low line scan rates. As a demonstration of how the SU-8 cantilevers enable this feature, large areas of a Celgard sample were imaged.
Celgard is a standard sample for assessing the speed performance of AFM imaging due to the challenge of tracking the freely-suspended fibrils in the material. On a standard MultiMode AFM system using large SU-8 cantilevers, we found that we could image Celgard with acceptable quality at line scan rates of 10 Hz, corresponding to a tip velocity of 100 μm/s. At this scan rate, the large silicon cantilever tracked very poorly and was unable to resolve the fibrils at all. On an unmodified commercial high-speed AFM system (FastScan, Bruker Nano Surfaces) we used our small SU-8 cantilevers to scan a 30×30 μm area of Celgard at a line scan rate of 4 Hz at 8192×3200 pixels, corresponding to a surface speed of 261 μm/s (FIG. 3b upper left image).
In comparison, the highest previously reported surface scan speeds we were able to find regarding Celgard using standard small AFM cantilevers was 56 μm/s. The upper right part of FIG. 3b presents a 2.4 μm digital zoom of the overall image corresponding to 656×256 pixels, showing the individual fibrils of the Celgard are still well resolved. The amplitude error and phase images in FIG. 3b show further evidence that the AFM tracks the surface well.
Electronic Readout of AFM Cantilevers
The optical beam detection technique is the most common method to detect the deflection of AFM cantilevers. It is easily implemented and very sensitive, however it requires a number of components, such as lasers, objectives, and photodetectors, which require both space and the ability to align them with the cantilever. Furthermore, optical diffraction sets the minimum width of the cantilever to a practical lower limit of about 2 μm. For these reasons, there has been much interest in the development of cantilevers with integrated deflection sensing elements (so-called self-sensing cantilevers). Cantilevers using resistive or piezoresistive, piezoelectric, thermal and capacitive detection techniques have been developed. Thus far, the performance of the current self-sensing cantilevers still lags behind the optical beam detection performance, and so these cantilevers are generally only used in situations where having optical beam detection is not possible.